Why does a turbocharged car only produce significant boost whilst the engine is under load. Is is something to do with how much exhaust is being produced or is there some kind of valve mechanism that controls this ??

i.e. running the engine at 3000rpm stood still would not produce as much boost as the engine running at 3000rpm whilst pulling the car uphill in gear.

4 Answers
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Let me lead off with an excellent book on the topic: Corky Bell's Maximum Boost. There is a sound treatment of the basics of turbocharger operation in addition to some dated and esoteric applications that are still interesting. For example, I find the discussion of turbocharging different types of carburetor to be of intellectual if not practical interest.

To summarize the points called out in your question, here are some of the major aspects of the turbocharged engine that are of interest:

The air flow: remember that an internal combustion engine is effectively an air pump. If we're talking about an engine that is operating "under load", we can assume that you've opened the throttle. For example, when rolling down hill, you don't need to hit the throttle so the entire intake to exhaust path is pumping a smaller air mass. However, driving uphill will require you to open the throttle (give it gas), adding air to the intake. This causes the engine computer to add fuel to the mixture. The fuel-air mixture is burned to produce energy. The exhaust from this combustion then proceeds to ...

The turbine: this is part that looks like the front of a jet engine sitting in the exhaust gas path. The turbine sits on one end of a spinning shaft. On the other side is the compressor. That is the part that actually makes the boost on the intake side of the engine. The more exhaust gas pushes past the turbine, the more it wants to spin and make boost on the compressor side. However, there is also ...

The wastegate: this is a valve that also sits in the exhaust gas path. It provides a short cut for the exhaust if the engine does not really need boost at this time. This can be used for peak boost control (too much boost can physically destroy your engine). This can be a purely mechanical sprung valve that stays closed up to a certain positive pressure in the intake path and then progressively opens as boost increases. It could also be under the direct control of the engine computer. For example, my car (in stock tune) was very annoying in its refusal to stay at peak boost in third gear. It also would refuse to boost past a certain point with partial throttle. The engine computer was effectively saying "no, that's enough fun for now."

For example, if I am rolling downhill in gear with my foot off the gas, the throttle is closed. There isn't enough air mass passing through the intake to exhaust path to make the turbo spin, wastegate or not.

However, the scene changes at the bottom of the hill as we climb the next rise. I have to open the throttle to get up the hill. If I'm in low gear, the RPMs will be higher, the exhaust gas energy will be higher and the turbine will spin right up. However, since I'd require partial throttle at lower gear for the same acceleration, my engine computer might veto boost past a certain point, opening the wastegate.

If I'm in high gear, the RPMs will be lower and I'll have to open the throttle wide open to get up the hill. However, the exhaust gas volume and velocity will be low and it's possible that I won't have enough energy for the turbo to make any significant positive pressure (e.g., around 40 mph in fifth in my car). Even though I'd really like to make boost in this situation, I won't be able to.

Okay so the wastegate control is electronic. What about in a car with a wastegate controlled by vacuum pressure. How does the car know that the engine is under load or not under load in order to open or close the wastegate ??
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general exceptionJan 25 '12 at 13:41

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@generalexception, you just answered your own question: if the engine is under vacuum, it has little load. The throttle must be nearly closed, reducing the total air mass through the system. As the throttle opens, the turbo will spin up as the exhaust energy increases. In a purely mechanical system, the wastegate will then only begin to open when the boost passes the threshold of the wastegate spring. In this case, the wastegate acts only as a boost ceiling control. Note edits of original comment: too many assumptions lead to something that read as factually wrong.
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Bob Cross♦Jan 25 '12 at 13:47

Hmmm, that's not how mine works... The wastegate on my car has an 8.5psi spring. It's normally closed and only opens when the pressure on the control line forces it open. Upstream from that is a boost controller diverter that bleeds off pressure, so the ECU can allow higher boost pressures (but never lower) by decreasing the pressure seen by the wastegate. The only reason there is lower boost is due to the lack of enough air mass to put the turbo into efficiency range and cause it to pressurize the intake. Rolling downhill, the throttle plate is closed, forcing vacuum, no mass=no boost.
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Brian KnoblauchMay 9 '13 at 11:49

@BrianKnoblauch, I'm rereading what I wrote and I agree that I conflated enough points that it's confusing for a novice reader. For example, I'm mushing in aspects of my car (in stock tune) without being explicit that it's more complicated than just boost <-> vacuum. I'm going to rewrite the answer.
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Bob Cross♦May 10 '13 at 12:50

@BrianKnoblauch, I've rewritten both the answer and the comment. Thanks for pointing out that I'd made a complete donkey out of myself!
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Bob Cross♦May 10 '13 at 13:23

In summary, turbocharger operates by having two turbines connected to the same rotating axis. One turbine is spun by the exhaust gases, which cause the other turbine to spin. The second one is what forces the air into the intake of the engine.

At idle rpm, there's barely any exhaust to produce boost. Open throttle => more air moved through the engine => more exhaust => more boost.

Amended question as you may have misunderstood it.
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general exceptionJan 25 '12 at 13:13

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@generalexception: read your amendment and yeah I did answer the question. It all comes down to how much air you are moving through the engine. If your car is sitting still, it takes VERY LITTLE open throttle to rev engine to 3k rpm. On the other hand, putting the car under any kind of load (i.e. moving uphill or acceleration...) will require much more open throttle and get to same RPM. More open throttle X rpm ==> more boost.
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DXMJan 25 '12 at 19:23

Not exactly... Remember, not all turbocharged engines use wastegate's.

What happens is say your engine is turning at 2000 RPM's with no load, now you put on a load, the RPM's drop and to bring it back up to 2000 RPM's you have to add throttle which is dumping fuel in to the engine. As you dump fuel, your increasing the combustion pressure, and ultimately the higher exhaust pressure's are going to spin the turbine fast, and produce more boost which will increase the combustion pressure even more (more O2 now available). See, on a engine with no load, even without a wastegate the turbo is not doing much.

and to make things a bit more complicated here, on a diesel engine it works similar but different. There is no intake air regulation on a diesel, the intake is always unrestricted and output is determined by the amount of fuel injected. This is why when diesel's rev they put out a lot of smoke until the engine catches up. Turbo charged diesel's rely on the turbocharger itself as a form of intake air regulation.

and as DXM pointed out, he is right. Open throttle => more air moved through the engine => more exhaust => more boost.
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FJSchrankJrMay 8 '13 at 14:13

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To be technically correct, opening the throttle in a standard gasoline engine allows more air into the system. The ECU sees that and adds more fuel to match. Then, that combination results in more output, which means more flow, etc... But correct, in that without the larger mass of air that an engine without a wastegate will not see any boost (example of a non-wastegate equipped turbo engine is used in the Piper Turbo Arrow. The throttle is pushed forward a little at a time and the manifold pressure watched. Throttle must be adjusted until right overall pressure achieved before takeoff).
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Brian KnoblauchMay 9 '13 at 11:54

Your car's engine uses more fuel to spin the engine to 3000RPM when it is under load as opposed to being revved in neutral. That's the short answer.

More fuel means more exhaust gas, which means more boost. Conversely, when you are revving in neutral, much less fuel is used, and therefore much less exhaust gas to spin the turbo. This is also why your car is heavier on gas going uphill than downhill.